Probabilities

Statistics 100A Homework 5 Solutions

Ryan Rosario

Chapter 5 1. Let X be a random variable with probability density function c(1 − x2 ) −1 < x < 1 0 otherwise

∞

f (x) = (a) What is the value of c?

We know that for f (x) to be a probability distribution −∞ f (x)dx = 1. We integrate f (x) with respect to x, set the result equal to 1 and solve for c.

1

1 =

−1

c(1 − x2 )dx cx − c x3 3

1 −1

= = = = c = Thus, c =

3 4

c c − −c + c− 3 3 2c −2c − 3 3 4c 3 3 4 .

(b) What is the cumulative distribution function of X? We want to find F (x). To do that, integrate f (x) from the lower bound of the domain on which f (x) = 0 to x so we will get an expression in terms of x.

x

F (x) =

−1

c(1 − x2 )dx cx − cx3 3

x −1

=

But recall that c = 3 . 4 3 1 3 1 = x− x + 4 4 2 =

3 4

x−

x3 3

+

2 3

−1 < x < 1 elsewhere

0

1

4. The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by,

10 x2

f (x) = (a) Find P (X > 20).

0

x > 10 x ≤ 10

There are two ways to solve this problem, and other problems like it. We note that the area we are interested in is bounded below by 20 and unbounded above. Thus,

∞

P (X > c) =

c

f (x)dx

Unlike in the discrete case, there is not really an advantage to using the complement, but you can of course do so. We could consider P (X > c) = 1 − P (X < c),

c

P (X > c) = 1 − P (X < c) = 1 −

−∞

f (x)dx

P (X > 20) =

10 dx x2 20 10 ∞ = − x 20 10 = lim − x→∞ x

1 2

∞

− −

1 2

=

(b) What is the cumulative distribution function of X? We want to find F (x). To do that, integrate f (x) from the lower bound of the domain on which f (x) = 0 to x so we will get an expression in terms of x. 10 dx 2 10 x 10 y = − x 10 10 = − − (−1) y 1− 0

10 y y

F (x) =

=

y > 10 y ≤ 10

2

(c) What is the probability that of 6 such types of devices at least 3 will function for at least 15 hours? What assumptions...